Nonmonotonic External Field Dependence of the Magnetization in a Finite Ising Model: Theory and MC Simulation

Using ${\varphi }^4$ field theory and Monte Carlo (MC) simulation we investigate the finite-size effects of the magnetization $M$ for the three-dimensional Ising model in a finite cubic geometry with periodic boundary conditions. The field theory with infinite cutoff gives a scaling form of the equation of state $h\mathrm{/}{M}^{\delta }=f(h{L}^{\beta \delta \mathrm{/}\nu },t\mathrm{/}{h}^{1\mathrm{/}\beta \delta })$ where $t=(T-T_c)\mathrm{/}{T}_c$ is the reduced temperature, $h$ is the external field and $L$ is the size of system. Below $T_c$ and at $T_c$ the theory predicts a nonmonotonic dependence of $f(x,y)$ with respect to $x \equiv hL^{\beta\delta/\nu}$atfixed$y \equiv t/h^{1/\beta \delta}$ and a crossover from nonmonotonic to monotonic behaviour when $y$ is further increased. These results are confirmed by MC simulation. The scaling function $f(x,y)$ obtained from the field theory is in good quantitative agreement with the finite-size MC data. Good agreement is also found for the bulk value $f(\mathrm{\infty },0)$ at $T_c$.